*> \brief \b DLAED9 used by sstedc. Finds the roots of the secular equation and updates the eigenvectors. Used when the original matrix is dense.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed9.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W,
* S, LDS, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N
* DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ),
* $ W( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAED9 finds the roots of the secular equation, as defined by the
*> values in D, Z, and RHO, between KSTART and KSTOP. It makes the
*> appropriate calls to DLAED4 and then stores the new matrix of
*> eigenvectors for use in calculating the next level of Z vectors.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The number of terms in the rational function to be solved by
*> DLAED4. K >= 0.
*> \endverbatim
*>
*> \param[in] KSTART
*> \verbatim
*> KSTART is INTEGER
*> \endverbatim
*>
*> \param[in] KSTOP
*> \verbatim
*> KSTOP is INTEGER
*> The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
*> are to be computed. 1 <= KSTART <= KSTOP <= K.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of rows and columns in the Q matrix.
*> N >= K (delation may result in N > K).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> D(I) contains the updated eigenvalues
*> for KSTART <= I <= KSTOP.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ,N)
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= max( 1, N ).
*> \endverbatim
*>
*> \param[in] RHO
*> \verbatim
*> RHO is DOUBLE PRECISION
*> The value of the parameter in the rank one update equation.
*> RHO >= 0 required.
*> \endverbatim
*>
*> \param[in] DLAMDA
*> \verbatim
*> DLAMDA is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the old roots
*> of the deflated updating problem. These are the poles
*> of the secular equation.
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (K)
*> The first K elements of this array contain the components
*> of the deflation-adjusted updating vector.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION array, dimension (LDS, K)
*> Will contain the eigenvectors of the repaired matrix which
*> will be stored for subsequent Z vector calculation and
*> multiplied by the previously accumulated eigenvectors
*> to update the system.
*> \endverbatim
*>
*> \param[in] LDS
*> \verbatim
*> LDS is INTEGER
*> The leading dimension of S. LDS >= max( 1, K ).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if INFO = 1, an eigenvalue did not converge
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Jeff Rutter, Computer Science Division, University of California
*> at Berkeley, USA
*
* =====================================================================
SUBROUTINE DLAED9( K, KSTART, KSTOP, N, D, Q, LDQ, RHO, DLAMDA, W,
$ S, LDS, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER INFO, K, KSTART, KSTOP, LDQ, LDS, N
DOUBLE PRECISION RHO
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), DLAMDA( * ), Q( LDQ, * ), S( LDS, * ),
$ W( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
DOUBLE PRECISION TEMP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3, DNRM2
EXTERNAL DLAMC3, DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DLAED4, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, SIGN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
*
IF( K.LT.0 ) THEN
INFO = -1
ELSE IF( KSTART.LT.1 .OR. KSTART.GT.MAX( 1, K ) ) THEN
INFO = -2
ELSE IF( MAX( 1, KSTOP ).LT.KSTART .OR. KSTOP.GT.MAX( 1, K ) )
$ THEN
INFO = -3
ELSE IF( N.LT.K ) THEN
INFO = -4
ELSE IF( LDQ.LT.MAX( 1, K ) ) THEN
INFO = -7
ELSE IF( LDS.LT.MAX( 1, K ) ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAED9', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( K.EQ.0 )
$ RETURN
*
* Modify values DLAMDA(i) to make sure all DLAMDA(i)-DLAMDA(j) can
* be computed with high relative accuracy (barring over/underflow).
* This is a problem on machines without a guard digit in
* add/subtract (Cray XMP, Cray YMP, Cray C 90 and Cray 2).
* The following code replaces DLAMDA(I) by 2*DLAMDA(I)-DLAMDA(I),
* which on any of these machines zeros out the bottommost
* bit of DLAMDA(I) if it is 1; this makes the subsequent
* subtractions DLAMDA(I)-DLAMDA(J) unproblematic when cancellation
* occurs. On binary machines with a guard digit (almost all
* machines) it does not change DLAMDA(I) at all. On hexadecimal
* and decimal machines with a guard digit, it slightly
* changes the bottommost bits of DLAMDA(I). It does not account
* for hexadecimal or decimal machines without guard digits
* (we know of none). We use a subroutine call to compute
* 2*DLAMBDA(I) to prevent optimizing compilers from eliminating
* this code.
*
DO 10 I = 1, N
DLAMDA( I ) = DLAMC3( DLAMDA( I ), DLAMDA( I ) ) - DLAMDA( I )
10 CONTINUE
*
DO 20 J = KSTART, KSTOP
CALL DLAED4( K, J, DLAMDA, W, Q( 1, J ), RHO, D( J ), INFO )
*
* If the zero finder fails, the computation is terminated.
*
IF( INFO.NE.0 )
$ GO TO 120
20 CONTINUE
*
IF( K.EQ.1 .OR. K.EQ.2 ) THEN
DO 40 I = 1, K
DO 30 J = 1, K
S( J, I ) = Q( J, I )
30 CONTINUE
40 CONTINUE
GO TO 120
END IF
*
* Compute updated W.
*
CALL DCOPY( K, W, 1, S, 1 )
*
* Initialize W(I) = Q(I,I)
*
CALL DCOPY( K, Q, LDQ+1, W, 1 )
DO 70 J = 1, K
DO 50 I = 1, J - 1
W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
50 CONTINUE
DO 60 I = J + 1, K
W( I ) = W( I )*( Q( I, J ) / ( DLAMDA( I )-DLAMDA( J ) ) )
60 CONTINUE
70 CONTINUE
DO 80 I = 1, K
W( I ) = SIGN( SQRT( -W( I ) ), S( I, 1 ) )
80 CONTINUE
*
* Compute eigenvectors of the modified rank-1 modification.
*
DO 110 J = 1, K
DO 90 I = 1, K
Q( I, J ) = W( I ) / Q( I, J )
90 CONTINUE
TEMP = DNRM2( K, Q( 1, J ), 1 )
DO 100 I = 1, K
S( I, J ) = Q( I, J ) / TEMP
100 CONTINUE
110 CONTINUE
*
120 CONTINUE
RETURN
*
* End of DLAED9
*
END